SOLUTION: My son is learning factoring polynomials and he has a question that he is confused about. Would appreciate your help in clarifying this. During factoring we factor the negative

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Question 1178730: My son is learning factoring polynomials and he has a question that he is confused about. Would appreciate your help in clarifying this.
During factoring we factor the negative sign of the leading coefficient so why is that not done when we factor a difference of squares like 81 - x^2?
If written in standard form it would be - x^2 + 81. So why is this incorrect :
81-x^2
- x^2 + 81
- ( x^2 - 81)
- ( x + 9 ) ( x-9)

His teacher says the correct answer is ( 9 + x) ( 9-x) and NOT -(x+9)(x-9)
Would appreciate your help. Thank you.
Found 3 solutions by MathLover1, ikleyn, greenestamps:
Answer by MathLover1(20849)   (Show Source): You can put this solution on YOUR website!


........use a rule:
in your case and
so, you will have

Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
Formally,  81-x^2  can be presented / factored as


    81-x^2 = -(x+9)*(x-9).



Formally,  -(x+9)*(x-9)  is IDENTICALLY the same as  (9+x)*(9-x).


So, formally your son's answer is correct.


But  (9+x)*(9-x)  is  traditionally  considered as  "more simple"  comparing with  -(x+9)*(x-9).


/////////////


On these simple factoring formulas, see the lessons


1. The square of the sum formula is                 .

      For details and examples of applications of this formula see the lesson The square of the sum formula in this site.


2. The square of the difference formula is     .

      For details and examples of applications of this formula see the lesson The square of the difference formula in this site.


3. The difference of squares formula is       .

      For details and examples of applications of this formula see the lesson The difference of squares formula in this site.



Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


I'm seeing a number of similar questions on this forum about factoring polynomials with a negative leading coefficient. Here are my thoughts about this.

An internet search shows me that it is customary (?) to call the greatest common factor of a polynomial negative if the leading coefficient is negative; for example

instead of

Apparently your son has been taught this ("During factoring we factor the negative sign of the leading coefficient...").

My comment is that in fact the factoring of a polynomial is not the solution to a problem; it is only a step in solving a problem. Whether or not you include the negative as part of the GCF is of no importance at all. In working some kinds of problems, NOT including the negative as part of the GCF might make solving the whole problem easier.

What is REALLY troubling is that it appears your son is being taught that the negative of a leading coefficient should be factored out, but when he does that with this problem to factor 81-x^2 as -(x+9)(x-9), he is told that his answer is wrong.

OF COURSE a student just learning a subject will be confused if he is being told things like that!!

Unfortunately, what is going on here is simply BAD TEACHING.

Again, the factoring itself is not the end goal; it is a step in solving an actual problem. And for that purpose, and answer of -(x+9)(x-9) is every bit as good as (9+x)(9-x).
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